Q. Did Judah use the name Maccabee?
He himself would probably not have used it, despite a statement in the 1st Book of Maccabees 2:4 that assumes that Judah had this name before his military exploits. It is more likely that he called himself simply Yehudah ben Matitiyahu, Judah the son of Mattathias the kohen.
Any additional name which he used would have been a reference to his descent from Chashmon, which, according to Josephus (“Antiquities of the Jews”, 12:263), is the name of his great-great-grandfather, though the sages sometimes regarded Chashmon or Chashmonai as the father of Mattathias and sometimes used the names Chashmon(ai) and Mattathias interchangeably.
An alternative view links Chashmon with the village of Cheshmon, mentioned in Joshua 15:27. The name Chashmon leads to the title Hasmonean. Hasmonean (but not Maccabee) is the name used in rabbinic sources, e.g. Middot 1:6 and Shabbat 21b, but not in the Books of Maccabees, which use the name Maccabee.
Judah himself is not mentioned by name in rabbinic sources, probably in order to prevent a Judah-cult arising. A similar reason is behind the omission of Moses from the Haggadah, apart from one incidental scriptural quotation.